Plasma Transport

The performance of a fusion reactor is set by how slowly the confined plasma’s particles and heat leak away. The field that deals with this “leaking” is plasma transport. On this page we first grasp transport through the familiar image of diffusion, then move through classical transport, neoclassical transport, and turbulent transport, and finally look all the way to the H-mode, confinement scaling laws, and the research frontier.

Start with Intuition (High School)

If you drop a single drop of ink into a glass of water, it slowly spreads throughout the whole glass even without stirring. This is diffusion: a phenomenon in which countless molecules move while randomly colliding with one another, producing a net flow from where it is concentrated to where it is dilute. Plasma transport is essentially the same. The center of a confined high-temperature plasma is dense and hot while the outside is dilute and cold, so particles and heat gradually try to spread outward.

In fusion, we want to make this spreading as slow as possible. Confining a plasma with a magnetic field is like putting hot tea into a thermos bottle: if the thermos has poor insulation, the tea cools off quickly. The measure of how long a plasma can hold its heat is called the energy confinement time, and the smaller the transport, the longer this time becomes.

Here is one fact that goes against intuition. A charged particle in a magnetic field moves by winding helically around the field lines, so while it can travel well along the field lines, it should hardly move at all across them. Yet in real devices, heat escapes sideways far faster than theory predicts. The story of tracking down why it escapes faster than expected is the highlight of this page. The culprit turned out to be the fine eddies that the plasma itself generates, namely turbulence.

The motion of a charged particle winding around field lines is treated in detail on the separate particle motion page. Since it forms the foundation for understanding transport, reading it alongside this page will help.

Understand the Physics (Undergraduate)

The starting point for expressing diffusion mathematically is Fick’s law, $\Gamma = -D \nabla n$, which states that the particle flow (flux) is proportional to the concentration gradient. Here $\Gamma$ is the particle flow per unit area, $n$ is the particle density, $D$ is the diffusion coefficient, and $\nabla n$ is the spatial gradient of density. The minus sign expresses that the flow goes “from where it is dense to where it is dilute.” Understanding transport amounts to estimating how large this $D$ is.

The diffusion coefficient can be estimated with the random-walk idea as $D \sim (\Delta x)^2 / \tau$. Here $\Delta x$ is the distance stepped sideways in a single collision, and $\tau$ is the time between one collision and the next. It is intuitive to remember it as the square of one step’s stride divided by the time for one step.

First consider classical transport. This is the most basic transport, caused only by Coulomb collisions between particles. A charged particle in a magnetic field rotates around the field lines with a radius $\rho$ (the Larmor radius), so each time it collides it jumps sideways by roughly $\rho$. Therefore the stride is $\Delta x \sim \rho$, and the diffusion coefficient becomes about

$$D_{\text{classical}} \sim \rho^2 \nu$$

where $\nu$ is the collision frequency. The stronger the magnetic field, the smaller the Larmor radius $\rho$, so $D$ decreases in inverse proportion to the square of the magnetic field. This is a welcome conclusion: the stronger the field, the better the confinement. Unfortunately, though, the diffusion coefficient measured in experiments is one to two orders of magnitude larger than this classical theory predicts. This discrepancy is precisely the anomalous transport problem that long troubled fusion research.

Deepen the Theory (Graduate)

Efforts to bridge the gap between classical theory and experiment proceeded in two directions: one was to correctly incorporate the effects of the magnetic configuration, and the other was to incorporate turbulence. Let us look at each in turn.

Neoclassical Transport and Banana Orbits

The plasma in a tokamak or helical device is doughnut-shaped (a torus), so the magnetic field strength is strong on the inside (the side near the doughnut hole) and weak on the outside. Because of this non-uniformity, some particles moving along the field lines are reflected in the strong-field region and become trapped. These are called trapped particles, while particles that can circle all the way around without being reflected are called passing particles.

Because trapped particles drift while going back and forth along the field lines, they trace a banana-shaped orbit when viewed in cross section. This is the banana orbit. The width of the banana widens to roughly $q/\sqrt{\varepsilon}$ times the Larmor radius (where $q$ is the safety factor and $\varepsilon$ is the inverse aspect ratio). Since the stride $\Delta x$ becomes much larger than in classical transport, the diffusion coefficient grows correspondingly. This transport theory, which incorporates the geometry of the torus configuration, is called neoclassical transport. Neoclassical transport predicts diffusion about an order of magnitude larger than classical transport, but even so it still does not reach the experimental values.

An important byproduct of neoclassical theory is the bootstrap current. This is the prediction that when a pressure gradient is present, a toroidal current flows spontaneously from the motion of trapped particles, and in high-pressure plasmas it can carry a substantial fraction of the total plasma current. Because it can reduce the need for external current drive, it is extremely important for fusion reactors aiming at steady-state operation.

Anomalous Transport and Turbulence

The culprit behind the remaining large discrepancy was turbulence. A plasma has energy sources in the form of temperature and density gradients, and these drive microscopic instabilities so that fine eddies and waves are constantly raised. The eddies stir the plasma sideways and carry heat and particles far faster than Fickian diffusion. Representative driving sources include the ion temperature gradient mode (ITG mode), the trapped electron mode (TEM), and the electron temperature gradient mode (ETG mode). These microscopic instabilities are directly connected to the plasma instabilities page, which deals with more macroscopic instabilities.

The benchmark for estimating the diffusion coefficient of turbulent transport is gyro-Bohm diffusion. Taking the eddy size to be about the Larmor radius and the time scale to be the inverse of the drift frequency, it becomes about

$$D_{\text{gyro-Bohm}} \sim \frac{\rho}{a}\, \frac{T}{eB}$$

where $a$ is the minor radius of the device, $T$ is the temperature, and $B$ is the magnetic field. The $T/eB$ part is the same as Bohm diffusion, an older empirical rule, but the essential point is that it is multiplied by the small ratio $\rho/a$. This shows that making the device larger to reduce $\rho/a$ relatively improves confinement, and it is one of the reasons large devices are advantageous.

Another important property of turbulent transport is the critical gradient and “stiff transport.” When the temperature gradient exceeds a certain critical value, turbulence suddenly strengthens and transport jumps up. As a result, the temperature profile is kept as if pinned near the critical gradient. No matter how strongly you heat the center, the central temperature does not rise much; rather, it is the edge (pedestal) temperature that pushes up the overall temperature. This is a design consequence of great importance.

Transport Barriers and the H-mode

In 1982, the H-mode (high-confinement mode) was discovered on Germany’s ASDEX tokamak. When the heating power exceeds a certain threshold, a thin layer forms at the plasma edge, turbulence is suppressed, and confinement improves by roughly a factor of two. This layer in which turbulence is suppressed is called a transport barrier.

The main mechanism is turbulence suppression by radial electric field shear. When the electric field changes sharply in the radial direction, the plasma flows at shifted speeds (sheared flow), and the turbulent eddies are stretched and torn apart. When the speed of stretching by the shear (the shearing rate) exceeds the growth rate of the turbulence, the eddies are destroyed before they can grow, and transport falls. A barrier that forms at the edge is called an edge transport barrier, while a barrier that forms inside the plasma with a negative magnetic shear configuration or rotation shear is called an internal transport barrier (ITB).

In the H-mode, a layer with a steep pressure gradient, the pedestal, forms at the edge. The higher the pedestal, the more the overall performance is raised, so its height greatly influences reactor performance. On the other hand, when the pressure gradient reaches the MHD stability limit, edge localized modes (ELMs) erupt periodically, imposing momentary heat loads on the divertor. Suppressing these ELMs is an important challenge for ITER.

Confinement Scaling Laws

Because first-principles calculation of transport is difficult, actual machine design uses empirical formulas obtained by regressing data from many devices, called confinement scaling laws. Widely used as the H-mode standard is the IPB98(y,2) scaling, which expresses the energy confinement time $\tau_E$ as a product of powers of current, magnetic field, density, heating power, device size, and so on. The design of ITER is based on this scaling.

Characteristic is the dependence on the heating power $P$, which is about $\tau_E \propto P^{-0.69}$: the more you strengthen the heating, the shorter the confinement time becomes. This is called power degradation, and it precisely reflects the stiff nature of turbulent transport. It does not simply mean “if you heat a lot, it gets hot,” and here appears one of the difficulties of fusion design. For the conditions under which fusion itself becomes possible, see the Lawson criterion page.

Research Frontier (PhD)

The principal means of predicting turbulent transport from first principles is gyrokinetic simulation. It is a framework that averages over the Larmor rotation to reduce the six-dimensional kinetics to five dimensions, making it possible to solve microscopic turbulence with realistic computational resources, and codes such as GENE, GKV, GYRO, and GS2 are used around the world. Research is advancing that, by comparing against experiment, quantitatively unravels which instability governs the transport in which region.

Current major research themes include the following. First is turbulence suppression, the question of how to suppress turbulence. The mechanism by which spontaneous sheared flows called zonal flows self-regulate turbulence, and suppression through external flow and electric field control, are being investigated. Second is nonlocal transport and avalanche-like transport, which are not determined by the local gradient alone and are drawing attention as phenomena in which disturbances in distant regions propagate instantaneously. Third is research on models such as EPED that predict the pedestal structure, and on resonant magnetic perturbations (RMPs) that control ELMs. Furthermore, in recent years there have been active attempts to replace the enormous data of gyrokinetics with fast surrogate models using machine learning, connecting to real-time control and integrated modeling. These are the frontier heading toward the still-unsolved grand problem of “complete prediction and control of anomalous transport.”

Check Your Understanding

Q1. What everyday phenomenon is plasma transport most like?

A The way a magnet attracts iron B Diffusion, like ink dropped into a glass of water spreading out C The way water falls from a high place to a low place D The way light reflects off a mirror

Q2. What quantities determine the diffusion coefficient of classical transport?

A By the random-walk idea, D is set by the product of the square of the Larmor radius rho and the collision frequency nu B It is determined only by the product of temperature and density C It grows in proportion to the strength of the magnetic field D It is determined only by the size of the device

Q3. What is the orbit traced by trapped particles called, and why does it matter for transport?

A A circular orbit, because the stride equals the Larmor radius B A straight orbit, because particles escape straight along the field lines C A banana orbit, because the banana width is larger than the Larmor radius and widens the stride D A helical orbit, because the particles fly out of the device

Q4. What is the main cause of experimental values exceeding classical and neoclassical theory, and what is its rough magnitude?

A External field non-uniformity is the main cause, with a magnitude of Bohm diffusion itself B Turbulence from the plasma itself, that is anomalous transport, is the main cause, with a magnitude of gyro-Bohm diffusion C Gravitational settling is the main cause, with a magnitude of half the classical diffusion D Impurity influx from the wall is the main cause, with a magnitude equal to neoclassical diffusion

Q5. Why does confinement improve in the H-mode?

A Because the whole plasma cools uniformly and turbulence stops B Because the magnetic field is weakened to slow the motion of particles C Because a transport barrier forms at the edge that suppresses turbulence, and radial electric field shear tears the eddies apart D Because the heating power is lowered to eliminate the temperature gradient

Related Topics

• Particle Motion: Covers Larmor rotation and drifts in a magnetic field, and the motion of trapped particles. It is the foundation for understanding transport.
• Plasma Instabilities: Covers the microscopic instabilities that drive turbulence and the macroscopic instabilities caused by pressure gradients.
• Lawson Criterion: Covers how confinement time, temperature, and density affect whether fusion is achievable. It shows the significance of reducing transport.